Let $A$ be a commutative Noetherian ring with identity and $M,N$ two finitely generated $A$-modules.
How to show $\operatorname{Hom}_A(M,N)$ is a finitely generated $A$-module?
2026-03-28 20:07:49.1774728469
How to show $\operatorname{Hom}_A(M,N)$ is a finitely generated $A$-module?
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Consider the surjection $A^n \to M$. This induces an injection $Hom_A(M,N) \to Hom_A(A^n, N) = N^n$. So this is a submodule of a finitely generated module and by the Noetherian condition, we are done.